Write gxx22x5 as a product of two nonconstant polynomials in

Write g(x)=x^2+2x+5 as a product of two non-constant polynomials in Z6[x] Note: I don\'t think we can check if g(x) is irreducible by checking if there is any roots since Z6 is not a field.

Solution

IF g(x) has any factorisation of two non constant polynomials they must be of the form

(x-a)(x-b)=g(x)

Since, g has degree 2 so non constant factors must have degree 1

So,

g(x)=0 means x=a,x=b are solutions to g(x)=0

So we check if any x for 0,1,2,..,6 is a solution to g(x)=0

x=0 ,   g(0)=5 mod 6

x=1 ,   g(1)=1+2+5=8= mod 6

x=2 ,    g(2)=4+4+5=13=1 mod 6

x=3,   g(3)=9+6+5=20=2 mod 6

x=4,   g(4)=16+8+5=29=5 mod 6

x=5, g(5)=25+10+5=4 mod 6

Hence no solutions to g(x)=0 mod 6

Hence, g(x) is irreducible